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A291737 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2 - S^3.

%I A291737 #4 Sep 11 2017 20:05:22
%S A291737 1,2,5,11,25,54,121,267,591,1310,2899,6422,14218,31486,69722,154389,
%T A291737 341881,757050,1676405,3712200,8220236,18202762,40307892,89257156,
%U A291737 197649588,437672056,969173912,2146123007,4752340053,10523504828,23303078705,51601960101
%N A291737 p-INVERT of (1,0,1,0,0,0,0,...), where p(S) = 1 - S - S^2 - S^3.
%C A291737 Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
%C A291737 See A291728 for a guide to related sequences.
%H A291737 Clark Kimberling, <a href="/A291737/b291737.txt">Table of n, a(n) for n = 0..1000</a>
%H A291737 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 2, 2, 3, 1, 3, 0, 1)
%F A291737 G.f.: -(((1 + x^2) (1 - x + x^2) (1 + 2 x + 2 x^2 + x^3 + x^4))/(-1 + x +  x^2 + 2 x^3 + 2 x^4 + 3 x^5 + x^6 + 3 x^7 + x^9)).
%F A291737 a(n) = a(n-1) + a(n-2) + 2*a(n-3) + 2*a(n-4) + 3*a(n-5) + a(n-6) + 3*a(n-7) + a(n-9) for n >= 10.
%t A291737 z = 60; s = x + x^3; p = 1 - s - s^2 - s^3;
%t A291737 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A154272 *)
%t A291737 Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291737 *)
%Y A291737 Cf. A154272, A291728.
%K A291737 nonn,easy
%O A291737 0,2
%A A291737 _Clark Kimberling_, Sep 11 2017